Comparison of total quotient curvature

TL;DR

This paper establishes comparison theorems for total quotient curvature, demonstrating that Einstein metrics optimize this curvature and generalize classical volume comparison and rigidity results.

Contribution

It introduces new comparison theorems for total quotient curvature, extending volume and rigidity theorems to this curvature measure, with characterizations of equality cases.

Findings

Einstein metrics attain sharp bounds on total quotient curvature.

Point-wise bounds lead to integral inequalities for quotient curvature.

Characterization of equality cases in the comparison theorems.

Abstract

In this paper, we establish some comparison theorems for the total quotient curvature. Specifically, we examine the behavior of the functional with respect to the total quotient curvature and prove that the background Einstein metric achieves a sharp bound on the total quotient curvature. We prove that if the quotient curvature satisfies a point-wise lower (or upper) bound relative to the Einstein metric, then the corresponding integral inequality holds. Also we can show characterize the equality case. Our result generalizes the volume comparison theorem for scalar curvature and the rigidity results for $\sigma_k$-curvature.